Abstract

The lateral shifts (i.e., the spatial GH shifts) of reflected wave from a uniaxial anisotropic chiral metamaterial slab are predicted based on the stationary-phase method. The two cases of the uniaxial chiral slab are investigated in detail. It is found that the negative uniaxial chiral slab can be realized for both the two cases as the group refraction of the left circularly polarized (LCP) wave is negative. The GH shifts of the perpendicular component (Δsp) are similar for both of the cases, which are enhanced at close-to-grazing incidence. However, the GH shifts of the parallel component (Δpp) can be enhanced near the Brewster angle. For case (I), by introducing a different chirality parameter and the thickness of the chiral slab, the transition between the positively and negatively enhanced GH shifts of the parallel component (Δpp) can be realized. For case (II), the magnitude and the position of the enhancement of the positively enhanced GH shifts (Δpp) can be tunable by adjusting the chirality and the thickness of the chiral slab.

© 2017 Optical Society of America

1. Introduction

We know that Goos-Hӓnchen (GH) effect refers to the reflected beam experiences a lateral shift in the incident plane from the position predicted by geometrical optics, as a light beam is totally reflected from a planar interface [1]. Recently, the extensive investigations on the lateral shifts have been done due to the potential applications in optical devices. For instance, the enhancement and tunable of GH shifts can be realized in different materials or structures, including photonic crystals [2, 3], periodic structures [4], antiferromagnet [5], absorptive media [6–8], and so on. Apart from the constant spatial GH shifts mentioned before, the light beam also experiences an angular GH shift, which can be theoretically and experimentally observed via the weak measurement scheme [9, 10]. Meanwhile, with the help of the weak measurements, the spatial and angular GH shifts can be observed in total internal reflection (TIR) [11] or in partial reflection [12, 13], Moreover, the spatial and angular GH shifts can also be observed from a lossy surface [14] or in a graphene [15, 16].

On the other hand, since the emergence of left-handed materials (LHMs), which has brought about great opportunities and sophisticated pathways to manipulate light. Thus, the lateral shifts (i.e., the spatial GH shifts) associated with metamaterials with negative refraction [17–19] are of great interest. Moreover, as the important candidate of the metamaterials with negative refraction [20], the chiral metamaterials have received great attention from both theoretical and experimental views [21–24]. Correspondingly, the lateral shifts of the isotropic chiral metamaterials have been studied, which predicted the large positive and negative lateral shifts can be obtained from an isotropic chiral metamaterial [25–27]. On the other hand, the isotropic metamaterials are very difficult to be realized in practice, and the most of the artificial structures are actually anisotropic with respect to the directions of incident waves. Thus with the research going on, the studies of the lateral shifts have been done from the isotropic metamateirals to anisotropic metamaterials [28, 29]. For example, the beam shifting of a Gaussian beam has been analyzed for an anisotropic negative refractive slab in detail [30]. Meanwhile, the thickness and physical parameters of the anisotropic metamaterial slab, as well as the incident angle of the light beam, strongly affect the properties of the lateral shifts [31–33].

In addition, in the previous studies, the lateral shifts of the anisotropic chiral metamaterial have rarely been reported, though the spatial and angle transverse shifts for a Gaussian incident beam have been investigated in Ref [34]. And by investigating the refractive index of the uniaxial chiral media [35, 36], the results show that the anisotropic chiral metamaterials are quite easy to be realized artificially, since the condition to realize negative refraction can be quite loose. Moreover, the optical properties of the uniaxial chiral slab and the guided modes of the uniaxial chiral waveguide have also been studied [37, 38], which predicted the anisotropic chiral metamaterials have more potential applications in optical devices. Taking this into account, in this paper, we would like to investigate the GH shifts of the reflected beam from an anisotropic chiral metamaterial slab with absorption. Generally, the spatial and angle GH shifts occur simultaneously. But for a collimated incident beam, the spatial shifts are dominant, for a focused beam, the angle shifts are dominant. That is to say for a well collimated Gaussian beam, the angular deviation vanishes in the geometrical optics limit. Thus, for simplicity, we consider a monochromatic plane wave, and the spatial GH shifts (i.e., the lateral shifts) are investigated without considering the angle GH shifts. The influences of the parameters of the anisotropic chiral metamaterial will be discussed in detail. The modulation of the enhanced beam shift with positive or negative values and the transition between them can also be obtained.

2. Formulation

The configuration for the uniaxial anisotropic chiral metamaterial slab with the thickness d is shown in Fig. 1, which the optical axial is perpendicular to the interface. In general, time dependence exp(iωt) is applied and suppressed. The constitutive relations of the uniaxial chiral slab are defined as [39]

D=ε¯¯ε0E+iκ¯¯ε0μ0H,B=μ¯¯μ0Hiκ¯¯ε0μ0E,
where
ε¯¯=(εt000εt000εz),μ¯¯=(μt000μt000μz),κ¯¯=(00000000κ).
ε0 and μ0 are the permittivity and permeability in vacuum. εt (μt) and εz (μz) are the relative permittivity (permeability) of the uniaxial chiral medium perpendicular and parallel to the optical axial. κ is the chirality parameter and describes electromagnetic coupling.

 figure: Fig. 1

Fig. 1 Schematic diagram of a light beam propagating through the uniaxial anisotropic chiral slab placed in free space.

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Here, we assume that a polarized wave is incident at an angle θi upon the surface of a uniaxial chiral slab. The surface is parallel to the xy plane, and the normal of the interface is along the z direction. In the uniaxial chiral medium, there are two propagation modes: a right circularly polarized (RCP) wave with wave number kR and a left circularly polarized (LCP) wave with wave number kL, which are generated as [37]

kR(L)=k0εtμtcos2θR(L)+sin2θR(L)/AR(L),
with AR(L)=12(μzμt+εzεt)±14(μzμtεzεt)2+κ2εtμt, and + , - are corresponding to RCP and LCP waves in sign “±”. θR(L) is the refraction angle of RCP (LCP) wave, which also refers to the phase refraction angle. k0(ωε0μ0) is the wave number in vacuum. The wave numbers of the incident (ki), reflected (kr) and transmitted (kt) waves are satisfied kt=kr=ki=k0, θt=θr=θi, kzR(zL)=kR(L)cosθR(L). According to the boundary continuity condition (kyR=kyL=kiy), we have kRsinθR=kLsinθL=k0sinθi.

In a chiral material, an electric or magnetic excitation will produce both the electric and magnetic polarizations simultaneously. As a consequence, the reflected wave must be a combination of both perpendicular (s polarized, TE) and parallel (p polarized, TM) components in order to satisfy the boundary conditions, so does the transmitted wave. For the random polarized incident wave, by matching the boundary conditions at the interface z=0 andz=d, we will have the relations as the following.

(M11M12M21eikRzdM22eikRzdM21M22M11eikRzdM12eikRzdM31M32M41eikLzdM42eikLzdM41M42M31eikLzdM32eikLzd)(ErsErpEtsEtp)=(M21M22M11M12M41M42M31M32)(EisEip),
with
M11=(YzRYzL)εt(kzR+ωμtg0cosθi),M12=(YzRYzL)YzLμt(g0kzRωεtcosθi),M21=(YzRYzL)εt(kzRωμtg0cosθi),M22=(YzRYzL)YzLμt(g0kzR+ωεtcosθi),M31=(YzRYzL)εt(kzL+ωμtg0cosθi),M32=(YzRYzL)YzRμt(g0kzLωεtcosθi),M41=(YzRYzL)εt(kzLωμtg0cosθi),M42=(YzRYzL)YzRμt(g0kzL+ωεtcosθi),
where YzR(zL)=g0εtiκ(AR(L)εzεt).

According to the cross polarization in the chiral material, the reflected and transmitted coefficients contain the perpendicular and parallel components. Therefore, both the co-polarized and cross-polarized terms are involved in the coefficients. Then, we have

(M11M12M21eikRzdM22eikRzdM21M22M11eikRzdM12eikRzdM31M32M41eikLzdM42eikLzdM41M42M31eikLzdM32eikLzd)(RssRspRpsRppTssTspTpsTpp)=(M21M22M11M12M41M42M31M32).

In the above equation, Rab and Tab (ab=ss,ps,sp,pp) are the reflected and transmitted coefficients, a and b correspond to the polarized state of the reflected and incident wave, respectively. Meanwhile, a=b corresponds to the co-polarized coefficients, a1b corresponds to the cross-polarized coefficients. The arbitrary reflected coefficient can be written as Rab=|Rab|eiϕab, ϕab is the phase of the reflectance.

What's more, according to the stationary-phase approach, the lateral shift (the spatial GH shift) of the reflected beam can be defined as

Δab=ϕabkx=ϕabθiθikx=1kicosθiϕabθi.

3. Results and discussions

We are now in a position to present numerical results on the lateral shifts of the uniaxial chiral slab. For numerical calculation, we let μt=μz=1 for simplicity. There will exist four cases for the permittivity of the uniaxial chiral slab:(I) εt>0,εz>0; (II) εt>0,εz<0; (III) εt<0,εz>0; (IV) εt<0,εz<0. In the following discussion, a transverse magnetic (TM polarized) wave is injected into the interface of the uniaxial chiral slab with the angle of incidence θi.

First of all, the case (I) is investigated, and taking into account the dissipation of the actual chiral materials in the resonant frequency, the parameters are chosen as εt=3+0.01i, εz=2+0.01i, ω=2π×10GHz, and d = 10mm. Without loss of generality, we consider two types of chiral slab: (1) a weak chirality asκ<εzμz (κ = 0.1, 0.3, 0.5); (2) a strong chirality κ>εzμz (κ = 3, 4, 6). Generally, the wave vector and the Poynting vector are not parallel to each other in the anisotropic metamaterial. Therefore, it is quite necessary to introduce the group refraction angle θgR and θgL for the anisotropic chiral slab, which can be given as [35]

θgR(L)=arctan(tanθR(L)AR(L)).

Now let us inspect the phase and group refraction angles for the weak and strong chirality, as shown in Fig. 2. It can be seen that whatever the chirality is weak or strong, there is always no total reflection. For both of RCP and LCP waves, the phase refraction angles (θR>0,θL>0) are positive, so are the group refraction angles (θgR>0,θgL>0) as the chirality is very weak (κ = 0.1, 0.3, 0.5). It indicates that the uniaxial anisotropic chiral metamaterial possesses positive refraction index and only supports the forward wave for the weak chirality. When the chirality is strong (κ = 3, 4, 6), the phase and group refraction angles are both positive (θR>0,θgR>0) for RCP wave, thus forward wave is always supported. But for LCP wave, the phase and group refraction angles are respectively positive (θL>0) and negative (θgL<0), which indicates the uniaxial anisotropic chiral metamaterial possesses negative refraction index and can support the backward wave. Then, we conclude that the uniaxial anisotropic chiral slab can be positive or negative only as adjusting the chirality κ. From Fig. 2, we also find the chirality parameter has little effect on the phase refraction, but affects the group refraction much. Moreover, for the anisotropic chiral slab with negative refraction, which corresponds to the large chirality (κ = 3, 4, 6) as seen in Fig. 2(d), the group refraction angle of LCP wave will decrease with the increase of the chirality. It means that the increase of the chirality leads to the decrease of the negative refraction.

 figure: Fig. 2

Fig. 2 For case (I), the phase refraction angle of RCP (a) and LCP (b) waves, the group refraction angle of RCP (c) and LCP (d) waves as a function of the angle of incidence θi for various κ.

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Then, for this type of uniaxial anisotropic chiral metamaterial, the dependence of the reflectivity |R|2 (a, b) and the GH shifts Δ/λ (c, d) for both perpendicular (subscript sp) and parallel (subscript pp) components on the angle of incidence are shown in Fig. 3. It is easily found that there is a peak in each reflectivity curve for the perpendicular component, at which |Rsp|2 reaches the maximum, as seen in Fig. 3(a). As increasing the chirality, the peak of the reflectivity increases (or decreases) for the weak (or strong) chiral slab. Therefore, the Brewster angle does not exist for the perpendicular components. And from Fig. 3(c), the GH shifts will be greatly enhanced at close-to-grazing incidence, the enhancement of the GH shifts (Δsp/λ) can change from positive to negative as the chirality varies from weak to strong. In contrast, for the parallel component [see Fig. 3(b) and 3(d)], when chirality is very weak (κ = 0.1), there exist two dips of the reflectivity curve, at which the reflectivity reaches the minimum magnitude and can be close to zero. Therefore, the two corresponding angles of incidence can be defined as the Brewster angles, ie., θB145 and θB263.5. As a consequence, one expects that the GH shifts are enhanced near the two Brewster angles, which have two negative peaks as seen in Fig. 3(d). Meanwhile, increasing the chirality will lead to the transition of the enhancements from two negative peaks to a positive and a negative one. Here, we note that for the weak chirality, the Brewster angle corresponding to the large angle of incidence will turned to be pseudo-Brewster angle due to the increase of the minimum reflectivity, which is no longer close to zero as the chirality is relatively large (κ = 0.5). The enhanced GH shift corresponding to the large angle of incidence also decreases. When chirality becomes strong (κ = 3, 4, 6), one dip will be seen, i.e., there exists one Brewster angle. Therefore, the GH shifts can be enhanced near the Brewster angle with a positive peak. From Fig. 3, we conclude that the Brewster angle does not exist in the cross-polarized reflection but exist in the co-polarized reflection. Moreover, increasing the chirality will lead to the transition of the enhancements from two negative peaks to a positive and a negative peaks, eventually become a positive one.

 figure: Fig. 3

Fig. 3 For case (I), the reflectivity|Rsp|2, |Rpp|2 (a, b) and the GH shift Δsp/λ, Δpp/λ (c, d) as a function of the angle of incidence θi for various κ.

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Since the GH shifts of the co-polarized reflection component are more attractive than that of the cross-polarized reflection component, we aim at the case of co-polarized reflection for both weak and strong chirality. The influences of the thickness of the slab on the GH shifts of the parallel components are shown in Fig. 4. The corresponding reflectivity is shown in the insets of Fig. 4. According to Fig. 4(a), the GH shifts can be negatively enhanced at the two Brewster angles at d = 10mm for weak chirality, which has also been shown in Fig. 3(b). By increasing the thickness of the slab, the two negative enhancement of the GH shifts can vary to a positive one, then varies to a negative one again as the further increase of the thickness. While for large chirality, there is always a Brewster angle for different thickness of the slab, and the GH shift can be positively enhanced at the Brewster angle at d = 10mm. By increasing the thickness, the enhanced GH shift can change from positive to negative, then to positive again. From Fig. 4 we will find by adjusting the thickness of the slab, the transition between the enhanced positive and negative GH shifts can be realized.

 figure: Fig. 4

Fig. 4 For case (I), the GH shift Δpp/λ of weak chirality (a) and strong chirality (b) as a function of the angle of incidence θi for various thickness d. The insets are the corresponding reflectivity.

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Apart from the aforementioned uniaxial chiral slab of case (I), we also consider the case (II), the corresponding parameters are chosen as εt=3+0.01i, εz=4+0.01i. The others parameters are the same to those in the case (I). The phase and group refraction angles for the weak and strong chirality are shown in Fig. 5. It can be seen that whatever the chirality is weak or strong, the phase and group refraction angles are always positive (θR>0,θgR>0) for the RCP wave. It indicates that the uniaxial anisotropic chiral metamaterial always supports the forward RCP wave. But for the LCP wave, the phase refraction angles are always positive (θL>0), while the group refraction angles are always negative (θgL<0). Thus the backward wave can be supported for both the weak and strong chirality, and the uniaxial anisotropic chiral metamaterial possesses negative refraction index. It means that the realization of the negative chiral metamaterial is not affected by the chirality parameter for the case (II). Here, we note that the chirality parameter will affect the negative refractive index much more for the strong chirality, and the negative refractive index will decrease with the increasing of the chirality. Comparing Fig. 2 and Fig. 5, we conclude that the uniaxial anisotropic chiral slab can be negative as the chirality κ is strong for case (I), but for case (II) it can be negative whatever the chirality is weak or strong.

 figure: Fig. 5

Fig. 5 For case (II), the phase refraction angle of RCP (a) and LCP (b) waves, the group refraction angle of RCP (c) and LCP (d) waves as a function of the angle of incidence θi for various κ.

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Furthermore, the dependence of the reflectivity |R|2 (a, b) and the GH shifts Δ/λ (c, d) for both perpendicular (subscript sp) and parallel (subscript pp) components on the angle of incidence are discussed. For the perpendicular component, as shown in Fig. 6(a) and 6(c), there is a peak in each reflectivity curve and the GH shifts will be greatly enhanced at close-to-grazing incidence. By comparison, we find that the variation trends of the reflectivity and the GH shifts are the same as those in Fig. 3(a) and 3(c), which means that the influences of the chirality on the perpendicular component are the same for the two cases of the uniaxial chiral slab, i.e., case (I) and case (II). But for the parallel component, there always exists a dip of the reflectivity curve whatever the chirality is weak or strong, at which the reflectivity reaches the minimum magnitude and can be close to zero, the corresponding angle of incidence can be defined as the Brewster angle. As a consequence, the GH shifts are positively enhanced near the Brewster angle, as seen in Fig. 6(d). Meanwhile, with the increasing of chirality, the enhancement of the positive GH shifts decreases (or increases) for the weak (or strong) chirality. Moreover, the Brewster angle shifts to the large angle of incidence with increasing the chirality. As a result, the incident angle corresponding to the enhancement peak shifts to the large angle as the chirality increases. According to Fig. 6, the magnitude and the position of the enhancement of the GH shifts can be adjusted by changing the chirality parameter.

 figure: Fig. 6

Fig. 6 For case (II), the reflectivity |Rsp|2, |Rpp|2 (a, b) and the GH shift Δsp/λ, Δpp/λ (c, d) as a function of the angle of incidenceθi for various κ.

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In the end, to show the influences of the thickness of the slab on the GH shifts for the co-polarized reflection. The GH shifts of the parallel components for different thickness are shown in Fig. 7 for different chirality. We can see the GH shifts always have one enhanced peak with the angle of incidence whatever the chirality is weak or strong. Meanwhile, the position of the peak shows a right shift, i.e., the peak shifts to the large incident angle. As chirality is small (see Fig. 7(a)), the enhancement of the GH shifts decreases with the increase of the thickness. While the enhancement of the GH shifts decreases firstly and then increases with increasing the thickness as the chirality is strong, seen in Fig. 7(b).

 figure: Fig. 7

Fig. 7 For case (II), the GH shift Δpp/λ of weak chirality (a) and strong chirality (b) as a function of the angle of incidence θi for various thickness d. The insets are the corresponding reflectivity.

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4. Conclusion

In summary, we construct a uniaxial anisotropic chiral metamaterial slab with the optical axis perpendicular to the interface. An investigation on the GH shifts of reflected waves has been done by using the stationary-phase approach. Since there are four cases of electromagnetic parameters for the uniaxial chiral slab, we focus our discussion on the refraction angles and the GH shifts for the two cases of the uniaxial chiral slab, i.e., case (I) (εt>0,εz>0) and (II) (εt>0,εz<0). Our numerical results show that for both of the cases, the uniaxial chiral slab can support the backward wave, whose group refraction index of LCP wave is negative. For case (I), the negative uniaxial chiral slab can be realized as the chirality is strong. While, it can be obtained whatever the chirality is weak or strong for case (II). In addition, the variation of the GH shifts of the perpendicular component (Δsp) are the same for the two cases of the uniaxial chiral slab, which are both greatly enhanced at close-to-grazing incidence. It indicates that there does not exist the Brewster angle in the cross-polarized reflection, and the influence of the chirality on Δsp for case (I) and (II) are the same. Meanwhile, For the co-polarized reflection, the Brewster angles are always exist for the two cases, and the chirality will largely affect the GH shifts. For case (I), the GH shifts of the parallel component (Δpp) can be negatively or positively enhanced near the Brewster angles. By adjusting the chirality, the enhancements of the GH shifts can transit from two negative peaks to a positive one. For case (II), the GH shifts of the parallel component (Δpp) are always positively enhanced near the Brewster angle. By adjusting the chirality, the magnitude of the enhanced GH shifts will decrease, the corresponding position will shift to large incident angle. In the end, whatever the chirality is weak or strong, the transition between negative and positive GH shifts of the parallel component can be realized by adjusting the thickness of the chiral slab for case (I). For case (II), the GH shifts are always positively enhanced and the magnitude of the enhancement of the GH shifts can be tunable by the thickness of the chiral slab.

Some other comments are in the following. We know that the spatial and angle GH shifts occur simultaneously in general. And the angle GH shifts are dominant for a focused incident beam, and then we would like to investigate the angle GH shifts for a focused incident beam in our next work.

Funding

National Natural Science Foundation of China (NSFC) (11604163, 11447229, 61371057).

Acknowledgments

We are thankful for the financial support and the fruitful cooperation with all partners.

References and links

1. F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1(7-8), 333–346 (1947). [CrossRef]  

2. L. G. Wang and S. Y. Zhu, “Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals,” Opt. Lett. 31(1), 101–103 (2006). [CrossRef]   [PubMed]  

3. P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007). [CrossRef]  

4. M. Miri, A. Naqavi, A. Khavasi, K. Mehrany, S. Khorasani, and B. Rashidian, “Geometrical approach in physical understanding of the Goos-Haenchen shift in one- and two-dimensional periodic structures,” Opt. Lett. 33(24), 2940–2942 (2008). [CrossRef]   [PubMed]  

5. F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009). [CrossRef]  

6. L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005). [CrossRef]   [PubMed]  

7. J. B. Götte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos-Hänchen effect for metals and dielectrics,” Opt. Express 16(6), 3961–3969 (2008). [CrossRef]   [PubMed]  

8. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009). [CrossRef]   [PubMed]  

9. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009). [CrossRef]  

10. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010). [CrossRef]  

11. G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013). [CrossRef]   [PubMed]  

12. S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Lett. 39(21), 6229–6232 (2014). [CrossRef]   [PubMed]  

13. S. Goswami, S. Dhara, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Optimized weak measurements of Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Express 24(6), 6041–6051 (2016). [CrossRef]   [PubMed]  

14. A. Aiello, M. Merano, and J.P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801(R) (2009). [CrossRef]  

15. X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos-Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39(19), 5574–5577 (2014). [CrossRef]   [PubMed]  

16. S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017). [CrossRef]  

17. P. R. Berman, “Goos-Hӓnchen shift in negatively refractive media,” Phys. Rev. E 66(6), 0676031 (2002). [CrossRef]  

18. J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002). [CrossRef]  

19. A. Lakhtakia, “On planewave remittances and Goos-Hӓnchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23(1), 71–75 (2003). [CrossRef]  

20. J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004). [CrossRef]   [PubMed]  

21. C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007). [CrossRef]  

22. J.F. Zhou, J.F. Dong, N.B. Wang, T. Koschny, M. Kafesaki, and C.M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79, 121104 (2009). [CrossRef]  

23. S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009). [CrossRef]   [PubMed]  

24. J. Dong, J. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009). [CrossRef]   [PubMed]  

25. J. F. Dong and B. Liu, “Goos-Hӓnchen shift at the surface of the chiral negative refraction medium,” in Proceedings of the 2008 International workshop on metamaterials, Nanjing, China (Academic,2008), pp. 98–101. [CrossRef]  

26. W. T. Dong, L. Gao, and C. W. Qiu, “Goos-Hӓnchen shift at the surface of chiral negative refractive media,” Progress in Electromagnetics Research, PIER 104, 255–268 (2009). [CrossRef]  

27. Y. Y. Huang, W. T. Dong, L. Gao, and D. W. Qiu, “Large positive and negative lateral shifts near pseudo-Brewster dip on reflection from a chiral metamaterial slab,” Opt. Express 19(2), 1310–1323 (2011). [CrossRef]   [PubMed]  

28. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009). [CrossRef]   [PubMed]  

29. Y. Huang, B. Zhao, and L. Gao, “Goos-Hänchen shift of the reflected wave through an anisotropic metamaterial containing metal/dielectric nanocomposites,” J. Opt. Soc. Am. A 29(7), 1436–1444 (2012). [CrossRef]   [PubMed]  

30. H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E . 71(6), 066612 (2005). [CrossRef]   [PubMed]  

31. Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008). [CrossRef]  

32. Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99(6), 066114 (2006). [CrossRef]  

33. M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008). [CrossRef]  

34. G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013). [CrossRef]  

35. Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113104 (2006). [CrossRef]  

36. Q. Cheng and T. J. Cui, “Reflection and refraction properties of plane waves on the interface of uniaxially anisotropic chiral media,” J. Opt. Soc. Am. A 23(12), 3203–3207 (2006). [CrossRef]   [PubMed]  

37. J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Progress In Electromagnetics Research, PIER 127, 389–404 (2012). [CrossRef]  

38. M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral waveguide under DB-boundary conditions,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013). [CrossRef]  

39. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Academic, 1994).

References

  • View by:

  1. F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1(7-8), 333–346 (1947).
    [Crossref]
  2. L. G. Wang and S. Y. Zhu, “Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals,” Opt. Lett. 31(1), 101–103 (2006).
    [Crossref] [PubMed]
  3. P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
    [Crossref]
  4. M. Miri, A. Naqavi, A. Khavasi, K. Mehrany, S. Khorasani, and B. Rashidian, “Geometrical approach in physical understanding of the Goos-Haenchen shift in one- and two-dimensional periodic structures,” Opt. Lett. 33(24), 2940–2942 (2008).
    [Crossref] [PubMed]
  5. F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
    [Crossref]
  6. L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
    [Crossref] [PubMed]
  7. J. B. Götte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos-Hänchen effect for metals and dielectrics,” Opt. Express 16(6), 3961–3969 (2008).
    [Crossref] [PubMed]
  8. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
    [Crossref] [PubMed]
  9. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
    [Crossref]
  10. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
    [Crossref]
  11. G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
    [Crossref] [PubMed]
  12. S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Simultaneous weak value amplification of angular Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Lett. 39(21), 6229–6232 (2014).
    [Crossref] [PubMed]
  13. S. Goswami, S. Dhara, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, “Optimized weak measurements of Goos-Hänchen and Imbert-Fedorov shifts in partial reflection,” Opt. Express 24(6), 6041–6051 (2016).
    [Crossref] [PubMed]
  14. A. Aiello, M. Merano, and J.P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801(R) (2009).
    [Crossref]
  15. X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos-Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39(19), 5574–5577 (2014).
    [Crossref] [PubMed]
  16. S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
    [Crossref]
  17. P. R. Berman, “Goos-Hӓnchen shift in negatively refractive media,” Phys. Rev. E 66(6), 0676031 (2002).
    [Crossref]
  18. J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002).
    [Crossref]
  19. A. Lakhtakia, “On planewave remittances and Goos-Hӓnchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23(1), 71–75 (2003).
    [Crossref]
  20. J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004).
    [Crossref] [PubMed]
  21. C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
    [Crossref]
  22. J.F. Zhou, J.F. Dong, N.B. Wang, T. Koschny, M. Kafesaki, and C.M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79, 121104 (2009).
    [Crossref]
  23. S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
    [Crossref] [PubMed]
  24. J. Dong, J. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009).
    [Crossref] [PubMed]
  25. J. F. Dong and B. Liu, “Goos-Hӓnchen shift at the surface of the chiral negative refraction medium,” in Proceedings of the 2008 International workshop on metamaterials, Nanjing, China (Academic,2008), pp. 98–101.
    [Crossref]
  26. W. T. Dong, L. Gao, and C. W. Qiu, “Goos-Hӓnchen shift at the surface of chiral negative refractive media,” Progress in Electromagnetics Research, PIER 104, 255–268 (2009).
    [Crossref]
  27. Y. Y. Huang, W. T. Dong, L. Gao, and D. W. Qiu, “Large positive and negative lateral shifts near pseudo-Brewster dip on reflection from a chiral metamaterial slab,” Opt. Express 19(2), 1310–1323 (2011).
    [Crossref] [PubMed]
  28. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
    [Crossref] [PubMed]
  29. Y. Huang, B. Zhao, and L. Gao, “Goos-Hänchen shift of the reflected wave through an anisotropic metamaterial containing metal/dielectric nanocomposites,” J. Opt. Soc. Am. A 29(7), 1436–1444 (2012).
    [Crossref] [PubMed]
  30. H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
    [Crossref] [PubMed]
  31. Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008).
    [Crossref]
  32. Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99(6), 066114 (2006).
    [Crossref]
  33. M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008).
    [Crossref]
  34. G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
    [Crossref]
  35. Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113104 (2006).
    [Crossref]
  36. Q. Cheng and T. J. Cui, “Reflection and refraction properties of plane waves on the interface of uniaxially anisotropic chiral media,” J. Opt. Soc. Am. A 23(12), 3203–3207 (2006).
    [Crossref] [PubMed]
  37. J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Progress In Electromagnetics Research, PIER 127, 389–404 (2012).
    [Crossref]
  38. M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral waveguide under DB-boundary conditions,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013).
    [Crossref]
  39. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Academic, 1994).

2017 (1)

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

2016 (1)

2014 (2)

2013 (3)

G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
[Crossref] [PubMed]

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral waveguide under DB-boundary conditions,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013).
[Crossref]

2012 (2)

J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Progress In Electromagnetics Research, PIER 127, 389–404 (2012).
[Crossref]

Y. Huang, B. Zhao, and L. Gao, “Goos-Hänchen shift of the reflected wave through an anisotropic metamaterial containing metal/dielectric nanocomposites,” J. Opt. Soc. Am. A 29(7), 1436–1444 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

2009 (7)

F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
[Crossref]

B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
[Crossref] [PubMed]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
[Crossref] [PubMed]

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

J. Dong, J. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009).
[Crossref] [PubMed]

W. T. Dong, L. Gao, and C. W. Qiu, “Goos-Hӓnchen shift at the surface of chiral negative refractive media,” Progress in Electromagnetics Research, PIER 104, 255–268 (2009).
[Crossref]

2008 (4)

Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008).
[Crossref]

M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008).
[Crossref]

M. Miri, A. Naqavi, A. Khavasi, K. Mehrany, S. Khorasani, and B. Rashidian, “Geometrical approach in physical understanding of the Goos-Haenchen shift in one- and two-dimensional periodic structures,” Opt. Lett. 33(24), 2940–2942 (2008).
[Crossref] [PubMed]

J. B. Götte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos-Hänchen effect for metals and dielectrics,” Opt. Express 16(6), 3961–3969 (2008).
[Crossref] [PubMed]

2007 (2)

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

2006 (4)

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99(6), 066114 (2006).
[Crossref]

Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113104 (2006).
[Crossref]

Q. Cheng and T. J. Cui, “Reflection and refraction properties of plane waves on the interface of uniaxially anisotropic chiral media,” J. Opt. Soc. Am. A 23(12), 3203–3207 (2006).
[Crossref] [PubMed]

L. G. Wang and S. Y. Zhu, “Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals,” Opt. Lett. 31(1), 101–103 (2006).
[Crossref] [PubMed]

2005 (2)

L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
[Crossref] [PubMed]

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
[Crossref] [PubMed]

2004 (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004).
[Crossref] [PubMed]

2003 (1)

A. Lakhtakia, “On planewave remittances and Goos-Hӓnchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23(1), 71–75 (2003).
[Crossref]

2002 (2)

P. R. Berman, “Goos-Hӓnchen shift in negatively refractive media,” Phys. Rev. E 66(6), 0676031 (2002).
[Crossref]

J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002).
[Crossref]

1947 (1)

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

Aiello, A.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

J. B. Götte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos-Hänchen effect for metals and dielectrics,” Opt. Express 16(6), 3961–3969 (2008).
[Crossref] [PubMed]

Albuquerque, E. L.

F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
[Crossref]

Baqir, M. A.

M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral waveguide under DB-boundary conditions,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013).
[Crossref]

Berman, P. R.

P. R. Berman, “Goos-Hӓnchen shift in negatively refractive media,” Phys. Rev. E 66(6), 0676031 (2002).
[Crossref]

Cai, L.

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Chen, H.

Chen, R.

M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008).
[Crossref]

Chen, S. Z.

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Chen, X.

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

Chen, X. D.

Chen, Y. Y.

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

Cheng, M.

M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008).
[Crossref]

Cheng, Q.

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99(6), 066114 (2006).
[Crossref]

Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113104 (2006).
[Crossref]

Q. Cheng and T. J. Cui, “Reflection and refraction properties of plane waves on the interface of uniaxially anisotropic chiral media,” J. Opt. Soc. Am. A 23(12), 3203–3207 (2006).
[Crossref] [PubMed]

Choudhury, P. K.

M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral waveguide under DB-boundary conditions,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013).
[Crossref]

Cui, T. J.

Q. Cheng and T. J. Cui, “Reflection and refraction properties of plane waves on the interface of uniaxially anisotropic chiral media,” J. Opt. Soc. Am. A 23(12), 3203–3207 (2006).
[Crossref] [PubMed]

Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113104 (2006).
[Crossref]

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99(6), 066114 (2006).
[Crossref]

Da, H. X.

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
[Crossref] [PubMed]

da Costa, J. A. P.

F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
[Crossref]

Dhara, S.

Dong, J.

Dong, J. F.

J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Progress In Electromagnetics Research, PIER 127, 389–404 (2012).
[Crossref]

Dong, W. T.

Y. Y. Huang, W. T. Dong, L. Gao, and D. W. Qiu, “Large positive and negative lateral shifts near pseudo-Brewster dip on reflection from a chiral metamaterial slab,” Opt. Express 19(2), 1310–1323 (2011).
[Crossref] [PubMed]

W. T. Dong, L. Gao, and C. W. Qiu, “Goos-Hӓnchen shift at the surface of chiral negative refractive media,” Progress in Electromagnetics Research, PIER 104, 255–268 (2009).
[Crossref]

Dumelow, T.

F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
[Crossref]

Feng, S.

M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008).
[Crossref]

Gao, L.

Ghosh, N.

Goos, F.

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

Goswami, S.

Götte, J. B.

Hanchen, H.

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

Hermosa, N.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Hou, P.

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

Huang, Y.

Huang, Y. Y.

Jayaswal, G.

Khavasi, A.

Khorasani, S.

Kong, J. A.

J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002).
[Crossref]

Koschny, T.

Kraftmakher, G.

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
[Crossref] [PubMed]

Lakhtakia, A.

A. Lakhtakia, “On planewave remittances and Goos-Hӓnchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23(1), 71–75 (2003).
[Crossref]

Li, J.

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Progress In Electromagnetics Research, PIER 127, 389–404 (2012).
[Crossref]

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Li, L. W.

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

Li, X.

Li, Z. Y.

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
[Crossref] [PubMed]

Lima, F.

F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
[Crossref]

Liu, M. X.

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Liu, Z. B.

Lu, X.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Luo, H. L.

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Mao, H. M.

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

Mehrany, K.

Merano, M.

G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
[Crossref] [PubMed]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

Mi, C. Q.

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Miri, M.

Mistura, G.

Nandi, A.

Naqavi, A.

Pal, M.

Pan, T.

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

Panigrahi, P. K.

Park, Y. S.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Pendry, J. B.

J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004).
[Crossref] [PubMed]

Qiu, C. W.

W. T. Dong, L. Gao, and C. W. Qiu, “Goos-Hӓnchen shift at the surface of chiral negative refractive media,” Progress in Electromagnetics Research, PIER 104, 255–268 (2009).
[Crossref]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

Qiu, D. W.

Rashidian, B.

Shi, J. L.

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

Soukoulis, C.

Sun, J.

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

Tian, J. G.

van Exter, M. P.

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

Wang, C.

Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008).
[Crossref]

Wang, L. G.

Wang, P.

Wang, Q.

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

Wang, Z. P.

Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008).
[Crossref]

Wen, S. C.

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Woerdman, J. P.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

J. B. Götte, A. Aiello, and J. P. Woerdman, “Loss-induced transition of the Goos-Hänchen effect for metals and dielectrics,” Opt. Express 16(6), 3961–3969 (2008).
[Crossref] [PubMed]

Woerdoman, J. P.

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

Wu, B. K.

J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002).
[Crossref]

Xiao, Y. T.

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

Xing, F.

Xu, C.

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
[Crossref] [PubMed]

Xu, G. D.

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

Yao, H. Y.

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

Yeo, S. T.

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

Zhang, S.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Zhang, W.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Zhang, X.

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Zhang, Y.

J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002).
[Crossref]

Zhang, Z. H.

Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008).
[Crossref]

Zhao, B.

Zhou, J.

Zhu, S. Y.

Zouhdi, S.

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

Ann. Phys. (2)

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

G. D. Xu, Y. T. Xiao, J. Li, H. M. Mao, J. Sun, and T. Pan, “Imbert-Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media,” Ann. Phys. 335, 33–46 (2013).
[Crossref]

Appl. Phys. Lett. (2)

J. A. Kong, B. K. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. 80(12), 2084–2086 (2002).
[Crossref]

S. Z. Chen, C. Q. Mi, L. Cai, M. X. Liu, H. L. Luo, and S. C. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

Electromagnetics (1)

A. Lakhtakia, “On planewave remittances and Goos-Hӓnchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23(1), 71–75 (2003).
[Crossref]

Eur. Phys. J. D (1)

M. Cheng, R. Chen, and S. Feng, “Lateral shifts of an optical beam in an anisotropic metamaterial slab,” Eur. Phys. J. D 50(1), 81–85 (2008).
[Crossref]

J. Appl. Phys. (1)

Q. Cheng and T. J. Cui, “Lateral shifts of optical beams on the interface of anisotropic metamaterial,” J. Appl. Phys. 99(6), 066114 (2006).
[Crossref]

J. Electromagn. Waves Appl. (1)

M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral waveguide under DB-boundary conditions,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdoman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

Opt. Commun. (1)

Z. P. Wang, C. Wang, and Z. H. Zhang, “Goos-Hӓnchen shift of the uniaxially anisotropic left-handed material film with an arbitrary angle between the optical axis and the interface,” Opt. Commun. 281(11), 3019–3024 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (6)

Phys. Rev. A (2)

P. Hou, Y. Y. Chen, X. Chen, J. L. Shi, and Q. Wang, “Giant bistable shifts for one-dimensional nonlinear photonic crystals,” Phys. Rev. A 75(4), 045802 (2007).
[Crossref]

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010).
[Crossref]

Phys. Rev. B (3)

F. Lima, T. Dumelow, E. L. Albuquerque, and J. A. P. da Costa, “Power flow associated with the Goos-Hӓchen shift of a normally incident electromagnetic beam reflected off an antiferromagnet,” Phys. Rev. B 79(15), 155124 (2009).
[Crossref]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and S. T. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance and negative refraction,” Phys. Rev. B 75(15), 155120 (2007).
[Crossref]

Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113104 (2006).
[Crossref]

Phys. Rev. E (2)

H. X. Da, C. Xu, Z. Y. Li, and G. Kraftmakher, “Beam shifting of an anisotropic negative refractive medium,” Phys. Rev. E.  71(6), 066612 (2005).
[Crossref] [PubMed]

P. R. Berman, “Goos-Hӓnchen shift in negatively refractive media,” Phys. Rev. E 66(6), 0676031 (2002).
[Crossref]

Phys. Rev. Lett. (1)

S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[Crossref] [PubMed]

Progress in Electromagnetics Research, PIER (2)

W. T. Dong, L. Gao, and C. W. Qiu, “Goos-Hӓnchen shift at the surface of chiral negative refractive media,” Progress in Electromagnetics Research, PIER 104, 255–268 (2009).
[Crossref]

J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Progress In Electromagnetics Research, PIER 127, 389–404 (2012).
[Crossref]

Science (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004).
[Crossref] [PubMed]

Other (4)

J.F. Zhou, J.F. Dong, N.B. Wang, T. Koschny, M. Kafesaki, and C.M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79, 121104 (2009).
[Crossref]

J. F. Dong and B. Liu, “Goos-Hӓnchen shift at the surface of the chiral negative refraction medium,” in Proceedings of the 2008 International workshop on metamaterials, Nanjing, China (Academic,2008), pp. 98–101.
[Crossref]

A. Aiello, M. Merano, and J.P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801(R) (2009).
[Crossref]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Academic, 1994).

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of a light beam propagating through the uniaxial anisotropic chiral slab placed in free space.
Fig. 2
Fig. 2 For case (I), the phase refraction angle of RCP (a) and LCP (b) waves, the group refraction angle of RCP (c) and LCP (d) waves as a function of the angle of incidence θi for various κ.
Fig. 3
Fig. 3 For case (I), the reflectivity | R sp | 2 , | R pp | 2 (a, b) and the GH shift Δ sp /λ , Δ pp /λ (c, d) as a function of the angle of incidence θi for various κ.
Fig. 4
Fig. 4 For case (I), the GH shift Δ pp /λ of weak chirality (a) and strong chirality (b) as a function of the angle of incidence θi for various thickness d. The insets are the corresponding reflectivity.
Fig. 5
Fig. 5 For case (II), the phase refraction angle of RCP (a) and LCP (b) waves, the group refraction angle of RCP (c) and LCP (d) waves as a function of the angle of incidence θi for various κ.
Fig. 6
Fig. 6 For case (II), the reflectivity | R sp | 2 , | R pp | 2 (a, b) and the GH shift Δ sp /λ , Δ pp /λ (c, d) as a function of the angle of incidenceθi for various κ.
Fig. 7
Fig. 7 For case (II), the GH shift Δ pp /λ of weak chirality (a) and strong chirality (b) as a function of the angle of incidence θi for various thickness d. The insets are the corresponding reflectivity.

Equations (8)

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D= ε ¯ ¯ ε 0 E+i κ ¯ ¯ ε 0 μ 0 H, B= μ ¯ ¯ μ 0 Hi κ ¯ ¯ ε 0 μ 0 E,
ε ¯ ¯ =( ε t 0 0 0 ε t 0 0 0 ε z ), μ ¯ ¯ =( μ t 0 0 0 μ t 0 0 0 μ z ), κ ¯ ¯ =( 0 0 0 0 0 0 0 0 κ ).
k R( L ) = k 0 ε t μ t cos 2 θ R( L ) + sin 2 θ R( L ) / A R( L ) ,
( M 11 M 12 M 21 e i k Rz d M 22 e i k Rz d M 21 M 22 M 11 e i k Rz d M 12 e i k Rz d M 31 M 32 M 41 e i k Lz d M 42 e i k Lz d M 41 M 42 M 31 e i k Lz d M 32 e i k Lz d )( E rs E rp E ts E tp )=( M 21 M 22 M 11 M 12 M 41 M 42 M 31 M 32 )( E is E ip ),
M 11 =( Y zR Y zL ) ε t ( k zR +ω μ t g 0 cos θ i ), M 12 =( Y zR Y zL ) Y zL μ t ( g 0 k zR ω ε t cos θ i ), M 21 =( Y zR Y zL ) ε t ( k zR ω μ t g 0 cos θ i ), M 22 =( Y zR Y zL ) Y zL μ t ( g 0 k zR +ω ε t cos θ i ), M 31 =( Y zR Y zL ) ε t ( k zL +ω μ t g 0 cos θ i ), M 32 =( Y zR Y zL ) Y zR μ t ( g 0 k zL ω ε t cos θ i ), M 41 =( Y zR Y zL ) ε t ( k zL ω μ t g 0 cos θ i ), M 42 =( Y zR Y zL ) Y zR μ t ( g 0 k zL +ω ε t cos θ i ),
( M 11 M 12 M 21 e i k R z d M 22 e i k R z d M 21 M 22 M 11 e i k R z d M 12 e i k R z d M 31 M 32 M 41 e i k L z d M 42 e i k L z d M 41 M 42 M 31 e i k L z d M 32 e i k L z d ) ( R s s R s p R p s R p p T s s T s p T p s T p p ) = ( M 21 M 22 M 11 M 12 M 41 M 42 M 31 M 32 ) .
Δ a b = ϕ a b k x = ϕ a b θ i θ i k x = 1 k i cos θ i ϕ a b θ i .
θ g R ( L ) = arc tan ( tan θ R ( L ) A R ( L ) ) .

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